Problem 46 - College Algebra Essentials

Question

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{\frac{1}{4}} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The functions \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{\frac{1}{4}} x\) are inverses of each other. The graph of \(f(x)\) is decreasing and asymptotic to the \(x\)-axis. The graph of \(g(x)\) is decreasing and asymptotic to the \(y\)-axis. They intersect at \((1,0)\).

Step 1: Graph of \(f(x)=\left(\frac{1}{4}\right)^{x}\)

To graph the exponential function, first identify a few key points. For \(x=0\), \(f(x)=1\). As \(x\) increases, \(f(x)\) decreases and approaches 0, and as \(x\) decreases, \(f(x)\) increases without bound. The graph passes through \((0,1)\) and is asymptotic to the \(x\)-axis.

Step 2: Graph of \(g(x)=\log _{\frac{1}{4}} x\)

To graph the logarithmic function, note that \(g(x)=\log _{\frac{1}{4}} x\) is the inverse of \(f(x)\). Therefore, its graph is a reflection of the graph of \(f(x)\) in the line \(y=x\). The graph passes through \((1,0)\), approaches the \(y\)-axis as an asymptote as \(x\) decreases, and decreases without bound as \(x\) increases.

Step 3: Combine the two graphs

Draw the graphs of \(f(x)\) and \(g(x)\) on the same set of axes. The \(x\)-axis is a horizontal asymptote for \(f(x)\) and the \(y\)-axis is a vertical asymptote for \(g(x)\). The two graphs intersect at \((1,0)\).